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The difference between implicit and explicit extrapolation is best
understood through an example. Following Claerbout (1985),
let us consider, for instance, the diffusion (heat conduction) equation
of the form
 
(1) 
Here t denotes time, x is the space coordinate, T (x,t) is the
temperature, and a is the heat conductivity coefficient.
Equation (1) forms a wellposed boundaryvalue problem if
supplied with the initial condition
 
(2) 
and the appropriate boundary conditions. Our task is to build a
digital filter, which transforms a gridded temperature T from one
time level to another.
It helps to note that when the conductivity coefficient a is
constant and the space domain of the problem is infinite (or periodic)
in x, the problem can be solved in the wavenumber domain. Indeed,
after the Fourier transform over the variable x, equation
(1) transforms to the ordinary differential equation
 
(3) 
which has the explicit analytical solution
 
(4) 
where denotes the Fourier transform of T, and k stands
for the wavenumber. Therefore, the desired filter in the
wavenumber domain has the form
 
(5) 
where for simplicity the coefficient a is normalized for the time
step equal to 1.
Returning now to the timeandspace domain, we can approach the filter
construction problem by approximating the spacedomain response of
filter (5) in terms of the differential operators
, which can be approximated
by finite differences. An explicit approach would amount to
constructing a series expansion of the form
 
(6) 
and selecting the coefficients a_{j} to approximate equation
(5). For example, the threeterm Taylor series expansion
around the zero wavenumber yields
 
(7) 
The error of approximation (7) as a function of k
for two different values of a is shown in the left plot of Figure
1.
error
Figure 1 Errors of secondorder explicit and implicit
approximations for the heat extrapolation.
An implicit approach also approximates the ideal filter
(5), but with a rational approximation of the form
 
(8) 
One way of selecting the coefficients b_{i} and c_{i} is to apply an
appropriate Padé approximation Baker and GravesMorris (1981)^{}. For example the [2/2] Padé
approximation is
 
(9) 
This approximation corresponds to the famous CrankNicolson implicit
method Crank and Nicolson (1947). The error of approximation (9) as
a function of k for different values of a is shown in the right
plot of Figure 1. Not only is it significantly smaller
than the error of the sameorder explicit approximation, but it also
has a negative sign. It means that the highfrequency numerical noise
gets suppressed rather than amplified. In practice, this property
translates into a stable numerical extrapolation.
The second derivative operator k^{2} can be approximated in practice
by a digital filter. The most commonly used filter has the
Ztransform D_{2} (Z) = Z^{1} + 2  Z, and the Fourier transform
 
(10) 
Formula (10) approximates k^{2} well only for small
wavenumbers k. As shown in Appendix A, the implicit scheme allows
the accuracy of the secondderivative filter to be significantly
improved by a variation of the ``1/6th trick''
Claerbout (1985). The final form of the implicit
extrapolation filter is
 
(11) 
where is a numerical constant, found in Appendix A.
heat
Figure 2 Heat extrapolation with explicit
and implicit finitedifferent schemes. Explicit extrapolation
appears stable for a=2/3 (left plot) and unstable for a=4/3
(middle plot). Implicit interpolation is stable even for larger
values of a (right plot).
A numerical 1D example is shown in Figure 2. The initial
temperature distribution is given by a step function. The
discontinuity at the step gets smoothed with time by the heat
diffusion. The left plot shows the result of an explicit extrapolation
with a=2/3, which appears stable. The middle plot is an explicit
extrapolation with a=4/3, which shows a terribly unstable behavior:
the highfrequency numerical noise is amplified and dominates the
solution. The right plot shows a stable (though not perfectly
accurate) extrapolation with the implicit scheme for the larger value of
a=2.
The difference in stability between explicit and implicit schemes is
even more pronounced in the case of wave extrapolation . For
example, let us consider the ideal depth extrapolation filter in the
form of the phaseshift operator
Claerbout (1985); Gazdag (1978)
 
(12) 
where , is the time frequency, and v is the
seismic velocity (which may vary spatially); we assume for simplicity
that both the depth step and the space sampling
are normalized to 1. A simple implicit approximation
to filter (12) is
 
(13) 
where . We can see
that approximation (13) is again a pure phase shift
operator, only with a slightly different phase. For that reason, the
operator is unconditionally stable for all values of a: the total
wave energy from one depth level to another is preserved. Operator
(12) corresponds to the CrankNicolson scheme for the
45degree oneway wave equation Claerbout (1985). Its
phase error as a function of the dip angle for different values of a is shown in Figure
3.
phase
Figure 3 The phase error of the
implicit depth extrapolation with the CrankNicolson method.

 
The unconditional stability property is not achievable with the
explicit approach, though it is possible to increase the stability of
explicit operators by using relatively long filters
Hale (1991b); Holberg (1988).
Next: Spectral factorization and threedimensional
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Stanford Exploration Project
10/9/1997